The hot spots conjecture can be false: Some numerical examples

Abstract

The hot spots conjecture is only known to be true for special geometries. It can be shown numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10 −3 . Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

A new approach to the hot spots conjecture

“…Jerison and Nadirashvili proved it for a class of domains with two axes of symmetry [11]. On the other hand Burdzy and Werner [5] and Burdzy [4] constructed counterexamples given by certain multiply-connected domains, see also the numerical study [16]. It remains open whether the conjecture is true for all convex or even all simply connected domains.…”

An upper bound on the Hot Spots constant

“…Burdzy [10] later constructed a planar counterexample with one hole. Kleefeld [22] used high-precison numerics to numerically investigate examples of domains with one hole: Burdzytype counterexamples [10] seem to be robust. Kleefeld also constructs an explicit (numerical) example of a domain for which…”

Improved upper bounds for the Hot Spots constant of Lipschitz domains

“…A novel approach to proving that HS2 holds for a domain or a class of domains is through a Hot Spots constant, an idea recently introduced by Steinerberger in [Ste21]; see also [Kle21]. The basic idea for a fixed domain D is to examine the quotient sup x∈D ϕ 2 (x)/ sup x∈∂D ϕ 2 (x) for a Neumann Laplacian eigenfunction ϕ 2 corresponding to µ 2 .…”

“…We also know from any of the counterexamples mentioned above that C 2 > 1. Indeed, Kleefeld [Kle21] uses superconvergent numerical methods to show that C 2 ≥ 1 + 10 −3 , and he hints that this approach may also be applicable in R 3 . On the other hand, bounding C d from above is the main topic of the present paper.…”

“…More generally, HS2 has been shown to hold for bounded cylinders in R d whose cross sections can be arbitrary Lipschitz domains in R d−1 ; see [Kaw85,Corollary 2.15]. However, several planar counterexamples to HS2 have been constructed; see [BW99,BB00,Bur05,Kle21]. All of these counterexamples feature multiply connected domains and it is widely believed that HS2 and the Hot Spots conjecture hold on arbitrary convex domains in R d .…”